{"id":"e-a3c708e6ae942d89","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c77c01e985dfc77d","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"b6e6d9d90f3b1815f22a51af30af2f044a15897cdad6e55f06a7e216d827a6e6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13480,"certificate_compressed_size":2551,"conclusion_name":"shift_x_plus_one of basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-317113238d23620b","role":"sequence","name":"basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-c77c01e985dfc77d","claim":"one_signed_coefficients"},{"node_id":"n-c77c01e985dfc77d","claim":"hadamard_kernel"},{"node_id":"n-c77c01e985dfc77d","claim":"interlacing"},{"node_id":"n-c77c01e985dfc77d","claim":"real_rooted"}]}